My nine most vital maths questions

First post here. Would dearly like to have some answers for book I am writing on aesthetics and psychology. Can't find FAQ section, or anywhere else these questions are answered succinctly, so here they are. Feel free to give yes/no answers, links or even to contemptuously brush aside (although I'd love to know, if so, why these questions are stupid). Needless to say I am not a mathematician and would appreciate as much simplification as possible without distortion.

1. Is it true that the number that pi represents, being infinitely "long", cannot be imagined / accurately represented / exactly known; only the symbol or approximation can be?
2. Does it follow that it is impossible to imagine / accurately represent / exactly know the area of a circle?
3. Is this also true for numbers like e and i? - that all we can know for sure about the calculations based on them are very good approximations?
4. What calculations are based on these numbers? What branches of modern mathematics are based on pi, e and i? Could you give me a simple list (logarithms, calculus, set theory, etc) or does it pretty much amount to "all"?
5. Is there some way that pi, e, i are connected to either prime numbers or chaos theory?
6. Why are prime numbers useful or beautiful? I read an Oliver Sachs book where two autistic twins were thinking up and then "enjoying" (as if drinking fine wine) massive prime numbers. Any idea why would this be so?
7. When a proof of theorum is said to be "beautiful" what criteria does it satisfy? Efficiency is obviously one of them - succinctness. Use of startling analogy another (i.e. reasoning from quite a remote perspective to the one at hand)? What about harmonic arrangement (the formula in some way "feels" nice, like a Fibonacci series looks nice)?
8. Quantum mechanics is, in some ways, the science of the impossible to imagine - what elements of maths are used to deal with wavicles and time travelling particles and whatnot? Does pi, e, i, primes or chaos figure in any way?
9. Would you say that interesting maths tends to come from interesting mathematicians? Knee jerk response I suppose is "of course not". But do the lives of the most "creative" mathematicians look different to the lives of the less brilliant? I know Fermat, for example, led a dull life, but is there any indication that there was "something about him" that set him aside from others in some other way than mathematical skill?

First post here. Would dearly like to have some answers for book I am writing on aesthetics and psychology. Can't find FAQ section, or anywhere else these questions are answered succinctly, so here they are. Feel free to give yes/no answers, links or even to contemptuously brush aside (although I'd love to know, if so, why these questions are stupid). Needless to say I am not a mathematician and would appreciate as much simplification as possible without distortion.

1. Is it true that the number that pi represents, being infinitely "long", cannot be imagined / accurately represented / exactly known; only the symbol or approximation can be?
2. Does it follow that it is impossible to imagine / accurately represent / exactly know the area of a circle?

Depends on what you mean by "known".
Sure, the decimal representation of pi is infinitely long, and hence, cannot be written down on a piece of paper.
The same is true, however, of the decimal representation of 1/3.

In both cases however, algorithms are well known that allows you to calculate however many correct decimals of either number's decimal representation.

What more do you really want?

Besides, what's so special about decimal representations of numbers anyway?

Neither of this has much to do with approximations in general; pi is a very well defined number whose properties can be relied upon "exactly" in any proof it occurs; it is only its decimal representation which is infinite and not finite. That's really all worth mentioning.

Thank you Magic Castle - perhaps you can guide me to the precise pages of Gower's book (which I have in front of me) where my questions are answered (I also mailed my questions to professor Gowers - he seems quite friendly). Unfortunately I do not have the ability or inclination to "do the math" required to answer my own questions in great depth; this is why I have asked others, more knowledgeable, to give me, or guide me towards, something I can comprehend.

Arildno. Thank you. I realise this question enters a realm outside of maths proper; e.g. the meaning of numbers, none of which can be "known" - in that they do no represent anything that can be accurately imagined. Nevertheless the precise number pi, as far as I understand it, is in some way different (like other irrational numbers?) in that not only does the thing this number "represent" elude the brain, but the number itself does. Is this so? Why does this not mean that when I use pi to measure a circle the answer I get is not quite precise? I suppose by "algorithms" you mean there are other non-decimal ways of representing pi which somehow eliminate its infinitely complex nature?

Also, is there not - when it comes to imaginability - some fundamental difference between 1/3, which, although "infinite" repeats itself, and pi, which does not?

The reason I ask these questions is not in order to calculate anything, but to understand better the relationship between mathematics and the capacity to imagine. Numbers like pi, e, zero, infinity, i and phi seem to form the base of all mathematics. Although I am not a mathematician I believe I am capable of grasping what these numbers have in common with each other, and with other realms of human experience and thought.

Thank you. I have read this book [a VSI b Gowers]. It does not answer any of these questions.

Really? He discusses question 7 in some detail, doesn't he?

Few of your questions seem well posed, nor to have mathematical answers. You'd be far better off posting them in philosophy (of mathematics/science).

1,2,3 are all about numerical accuracy, which is not the same thing as accuracy in mathematics. You're mixing 'real life' (where mathematics is used) with mathematics itself. [itex]\pi[/itex] is a perfectly good and accurate symbol.

My two cents. Being much of an ignorant myself, I can volunteer to state some of the obvious, to no-one disgrace. : )

1) (The first will be longer, serving as an introduction.) Imagine you start with the integer numbers only. The moment you want to divide 3 by 4, you can't; you need to extend your numbers with a new concept, where the old is still possible but the new can also be done. A similar thing has been done, when starting from fractions and finding out can you can't extract the square root of 2.

The new constructions (in this case you can google for "Cauchy sequences", or "Dedekind cuts") are provided with mechanisms to determine when two of them are equivalent (and thus represent the same number), or how to add or multiply them, or how to put the old numbers in the new representation. Once this is done, they are just as numbers as the fractions are.

Keep in mind that mathematics is a realm of symbols: "53" is no less a symbol than pi, sparing you the hassle of bringing 53 stones and putting them in a line. For a mathematician, pi or e are as exact and accurate as 53. The integers themselves are symbols constructed from an starting point (1) and a succesor operation. As for their "reality", this is a vague subject (reality is best left to your imagination :). Try to produce a metal bar of exactly 1 meter, and you'll see that the problems with reality have nothing to do with the length being integer, fractional or irrational. Pi or e are only approximations when you compare them to fractions, which are an abstraction in themselves.

The imaginary i is a different kind of animal. When you can't extract the square root of -1, you extend the numbers by a different method than above, by representing them as pairs; the old numbers become (n, 0), and i is (0, 1), the base of the new (second) dimension of numbers of the form (0, n). As such, i is a Gaussian integer: it is just a pair of integers, in many senses much like the fraction 3/4 is also a pair of integers.

2) Thus the area of a circle is exactly pi times the radius squared, provided you have an exact expression for the radius itself.

3) Ditto.

4) This is best left to someone knowing better, but a neat example is Euler's formula for complex numbers, e^ix = cos x + i sin x, or if you prefer, the pair (e, 0) raised to the power of (0, x) gives you the pair (cos x, sin x); a special case of which says that e to the power "i times pi" equals exactly 1.

5) No idea. An answer from an specialist would be in order.

6) They are (or can be) a basement for the construction of numbers. Each integer has a unique set of prime factors; it's like all numbers spawn from the primes. There is an inherent beauty to order (I'm probably quoting Jung) and to simple, elegant relationships. Primes usually share that rank. The relationships mentioned in 4) does too.

7) My gut guess is that it provides a bridge where none existed before. And note that this has a relation to the shortness or succintness of it: if the lenght of the proof were a measure of the bridged distance, a short bridge makes the two subjects even closer. Also, an "elegance" component might be one of surprisingly bringing to relation a subject which seemed unrelated.

8) Peak ignorance here. My guess is that statistics play a major role.

9) Define "interesting". If an interesting mathematician is one that produces interesting math, then the statement is a tautology. If it's meant that the mathematician is an interesting person, then I'd say the two conditions are mostly unrelated, except perhaps when it comes to divulgation. Claims about mathematicians being dull, introverted or neurotic are probably of a projective nature. :) , and certainly not specific to mathematics, but common to any intense mental activity. Meaning, chess players are worse.

. . .

It is good to note also why these kind of threads usually raise little enthousiasm. Too often they are posed by people who would rather been given the understanding, instead of looking for it by themselves. Not that it is your case; but it hits a tiresome spot. Nothing wrong with asking for references to study, as long as you go and do your homework. (And come back after a few months.)

My apologies for any squeech of my non-native English. I hope this can be of help.

As for "occurrence" of the numbers e and pi in practice, the interesting relationship
[tex]\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}[/tex]
along with that integrand's close intimacy with the probability density function in the normal distribution case makes e and pi occur quite frequently in statistical calculations.

Pi can be adequately "imagined" as the ratio between a circle's circumference and its diameter. End of imagination, since we may prove there is a unique number having that particular property.

Thank you Arildno. I'm afraid I don't understand "since we may" in this sentence.

Really? He discusses question 7 in some detail, doesn't he?

Question 7 is in the only one. I asked it because I was hoping for some greater insight or other observations.

Few of your questions seem well posed, nor to have mathematical answers. You'd be far better off posting them in philosophy (of mathematics/science).

Thank you Mr Grime. I'm not quite sure what you mean by "well posed". I take it by "well" you are not referring to my hazy understanding of the concepts I am seeking to understand - of this, clearly, I am guilty. If, by "well" you mean a standard used by mathematicians, I am ignorant of it I'm afraid. I'd like to know what you mean though. As for posting in philosophy, you may be right. I posted here because I was hoping from answers by mathematicians and not philosophers.

1,2,3 are all about numerical accuracy, which is not the same thing as accuracy in mathematics. You're mixing 'real life' (where mathematics is used) with mathematics itself. Pi is a perfectly good and accurate symbol.

Yes, I am deliberately mixing "real life" (I think) with maths. I know that pi is a good and accurate in maths. I just can't see if or how it applies to "real life".

Turning to Dodo, for whose long patient reply I am particularly grateful, I'm afraid I still do not understand your answer to my first or second questions. I'm sorry.

2) Thus the area of a circle is exactly pi times the radius squared, provided you have an exact expression for the radius itself.

I realise that the area is exactly pi times the radius squared. But if the radius is, for example, one hundred kilometres; what does that make the area? Isn't the answer imprecise?

They are (or can be) a basement for the construction of numbers. Each integer has a unique set of prime factors; it's like all numbers spawn from the primes. There is an inherent beauty to order (I'm probably quoting Jung) and to simple, elegant relationships. Primes usually share that rank. The relationships mentioned in 4) does too.

How are primes "ordered"? Are they ordered in the same way as a cube, or as a logarithmic spiral - the latter being less beautiful than the former to most people's eyes.

My gut guess is that it provides a bridge where none existed before. And note that this has a relation to the shortness or succintness of it: if the lenght of the proof were a measure of the bridged distance, a short bridge makes the two subjects even closer. Also, an "elegance" component might be one of surprisingly bringing to relation a subject which seemed unrelated.

Yes, I see. That's clear. I wonder about the "nice feel" element though. Are some proofs elegant like a flower is, or somehow giving a feeling of balance or some such?

Define "interesting". If an interesting mathematician is one that produces interesting math, then the statement is a tautology. If it's meant that the mathematician is an interesting person, then I'd say the two conditions are mostly unrelated, except perhaps when it comes to divulgation. Claims about mathematicians being dull, introverted or neurotic are probably of a projective nature. :) , and certainly not specific to mathematics, but common to any intense mental activity. Meaning, chess players are worse.

Yes, I mean an interesting person, having character, flame, originality about him.

It is good to note also why these kind of threads usually raise little enthousiasm. Too often they are posed by people who would rather been given the understanding, instead of looking for it by themselves. Not that it is your case; but it hits a tiresome spot. Nothing wrong with asking for references to study, as long as you go and do your homework. (And come back after a few months.)

Quite. I would like references. But I am not seeking to work on mathematics; I have my hands full. I know from experience that nothing, no matter how complex, cannot be explained to a laymen so that he grasps it, and this is what I ask for. Specialists are prone to look down on those who do not share their vocabulary, and to hide behind the power that their knowledge gives them - in their world. Often they spend so long in this world that they even forget what it is like to live outside it.

An example might be when Arildno tells me

[tex]\int_{-\infty}^{\infty}e^{-x^{2}}dx=\sqrt{\pi}[/tex]

I have pointed out that I am not a mathematician, and so, while grateful for the time taken to post an answer, I cannot understand it. What is the long thing on the right, like a treble clef? What is d? What is x? What is an intrgrand? What is probability density function? What is the normal distribution case?

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

I know that pi is a good and accurate in maths. I just can't see if or how it applies to "real life".

Mathematics is not reponsible for how its members appear in real life. Math deals with abstract quantities and structures. Some of these things may map very well onto real-life situations, some not so well, and some not at all. But this no fault of the mathematics.

I realise that the area is exactly pi times the radius squared. But if the radius is, for example, one hundred kilometres; what does that make the area? Isn't the answer imprecise?

No, the area is (half) as precisely [itex]\pi*10000[/itex] sq. kilometers as the radius is 100 km. The imprecision comes from how well we know the radius, not from how well we know pi.

How are primes "ordered"? Are they ordered in the same way as a cube, or as a logarithmic spiral - the latter being less beautiful than the former to most people's eyes.

No, they are not ordered as either. In fact, they are not known to be ordered in any way that can be written down using only previously familiar functions.

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

Thank you Mr Grime. I don't think I've made "what I wish to know" clear for you. I am not seeking unambiguous answers for unambiguous questions, but interesting answers for (to me) interesting questions, which would seem to be something else entirely (although perhaps alien to some people).

Mathematics is not reponsible for how its members appear in real life. Math deals with abstract quantities and structures. Some of these things may map very well onto real-life situations, some not so well, and some not at all. But this no fault of the mathematics.

Thank you Goku. You use the word "fault" - although, of course, I am not blaming mathematics or mathematicians. I am simply enquiring as to the relationship between real life and maths.

No, the area is (half) as precisely pi x 10000 sq. kilometers as the radius is 100 km. The imprecision comes from how well we know the radius, not from how well we know pi.

I see. But what is the answer? Forgive my ignorance, please, but if I multiply 10000 by pi to three decimal places won't I get a different answer to multiplying 10000 by pi to three million decimal places?

No, they are not ordered as either. In fact, they are not known to be ordered in any way that can be written down using only previously familiar functions.

I see, thank you. But I asked how primes were ordered because Dodo told me that prime numbers were beautiful because "beauty comes from order". In other words I was referring to the order in primes, rather than the order of them. I am aware that no-one has yet found a way to predict their distribution. It is their apparent "beauty" that interests me.

If integers are molecules, then primes are atoms. As each molecule can be reduced to its component atoms, so each integer can be expressed as a unique product of primes.

This is an interesting way of looking at them Christian. Thank you. Are integers in some way defined as being able to be expressed by primes, or is it just a happy coincidence that they can be? How, do you think, is either fact related to their beauty, if at all?

Given what you know of mathematics, and what you wish to know of mathematics, I don't think answers in web forums are going to help you. I think you're going to need to do some serious reading in books.

Oh, and well-posed simply means 'can be answered because there is no ambiguity in the quesiton'.

Matt, off topic, but here is a different version of your quote in the sig, again, with no attribution.

http://cs.union.edu/~postowb/cookie.html [Broken]

Search for 'waste' in the page and you will find it.

Here is the same version in a different context, about three jokes down,

Wiener, Norbert (1894-1964)
The Advantage is that mathematics is a field in which one's blunders tend to show very clearly and can be corrected or erased with a stroke of the pencil. It is a field which has often been compared with chess, but differs from the latter in that it is only one's best moments that count and not one's worst. A single inattention may lose a chess game, whereas a single successful approach to a problem, among many which have been relegated to the wastebasket, will make a mathematician's reputation.
Ex-Prodigy: My Childhood and Youth.

I see. But what is the answer? Forgive my ignorance, please, but if I multiply 10000 by pi to three decimal places won't I get a different answer to multiplying 10000 by pi to three million decimal places?

You will, because you are multiplying by two different numbers.

pi to three decimal places does not equal pi to three million decimal places does not equal pi.

There is only one answer when you multiply pi by 10000, but there are infinite representations. If you want to see what that looks like, I would ask you how? I can show it to you in terms of square millimeters. Just give me a compass that can measure out 100 millimters, and I will draw you a circle. The space that that cicle takes up is one representation of pi*10000 (note that pi*10000 is yet another representation). I can't show it to you in terms of decimal representation.

But this makes it no less real (or I guess in your terms "imaginable") than the intetegers.

I can represent pi as a symbol [tex] \pi [/tex] one two and three don't have greek letters that can represent them. Symply because mathematicians have not defined them. Had we difined our whole universe and number system differently perhaps we would be able to represent the ratio of a circumference and a diameter a bit easier and the intergers might be impossible.

Thank you Diffy. So what is the most exact or imaginable answer to the question pi x 10000?

(By the way, I apologise to mathematicians reading this, to you, nonsense. It must be like Kasparov being asked why can't the building jump like the horsey. I should be asking these questions elsewhere, I know - but I've started so I'll plug on. All I can say, in begging your indulgence, is that I know there are some great footballers who enjoy kicking a ball around with a five year old).

Thank you Diffy. So what is the most exact or imaginable answer to the question pi x 10000?

I do not quite know what you mean. pi x 10000 is not a question, so it does not have an answer.The number pi x 10000 is pi x 10000, and nothing else, and therefore is most exact.

If you are asking about what an irrational (or real number in general) is, it is merely a partition of the rational numbers into two disjoint sets, such that all elements of one set is greater than all elements of the other, and such that the one with lesser elements has no greatest element. So essentially pi is one such partition of rationals. A decimal representation of a real number is a convenient way of representing real numbers, but it is merely a representation of the real number (i. e. it is a defined symbol and not the actual symbols of the real numbers themselves, and not the real number itself), and is by no means the only representation.